A Combined MMSE-ML Detection for a SpectrallyEfficient non Orthogonal FDM Signal

Ioannis Kanarast , Arsenia Chortit , M.R.D. Rodriguest , and Izzat Darwazeht

tTelecommunications Research Group, Dept. E&EE, University College LondonLondon, UK, Email: Lkanaras@ee.uc1.ac.uk.a.chorti@ee.uc1.ac.uk.Ldarwazeh@ee.uc1.ac.uk

tInstituto de Telecomunicacoes, Department of Computer Science, University of PortoPorto, Portugal, Email: mrodrigues@dcc.fc.up.pt

The proposed FDM receiver consists of two stages. Initially,the received signal is processed by a bank of N correlators thatextract a set of sufficient statistics. The correlation processesare performed with a set of orthonormal basis functions bn ( t),generated from the FDM carriers using the Gram Schmidt(GS) orthonormalisation method. Hence, the output of the kth

receiver correlator can be expressed as

Rk = iT r(t)b'k(t)dt, k = 0, ... N - 1 (3)

[4]. On the other hand, linear Zero Forcing (ZF) detectionexhibited unsatisfactory error performance.

Consequently, to tackle the detection problem, we investi-gate the performance of alternative techniques. In particular,we consider the linear Minimum Mean Squared Error (MMSE)detection. In addition, stimulated by relevant research resultsin Multiple-Input Multiple-Output (MIMO) systems area [5],[6], and [7], we introduce and evaluate a combined MMSE -ML scheme.

This paper is organised as follows: In sections II and III wedescribe the FDM system and alternative detection techniques.In section IV we provide simulation results concerning theerror performance and the complexity of the methods. Finally,we summarize our main conclusions, outlining future researchdirections.

II. FDM SYSTEM MODEL

The original FDM system architecture is described in [4].Initially, the input stream is split into N parallel streamsthat modulate N different carriers according to a modulationscheme of size M. The FDM carriers are separated in fre-quency by a distance ~f that is only a fraction of the inverseof the FDM symbol period T. Consequently, the bandwidth ofthe transmitted signal is reduced by a factor ~fT < 1. Thecomplex envelope of the transmitted FDM symbol is given by

N-l

s (t) = Jr L SneJ27rn6.jt, t E [0, T] , (1)n=Q

where Sn is the modulation symbol for the nth carrier.For channels impaired by Additive White Gaussian Noise

(AWGN), n(t), we can write the received signal r(t) as

Abstract-In this paper, we investigate the possibility of reliableand computationally efficient detection for spectrally efficientnon-orthogonal Multiplexing (FDM) system, exhibiting varyinglevels of intercarrier interference. Optimum detection is basedon the Maximum Likelihood (ML) principle. However, ML isimpractical due to its computational complexity. On the otherhand, linear detection techniques such as Zero Forcing (ZF)and Minimum Mean Square Error (MMSE) exhibit poor per-formance. Consequently, we explore the combination of MMSEestimation with ML estimation around a neighborhood of theMMSE estimate. We evaluate the performance of the differ-ent schemes in Additive White Gaussian Noise (AWGN), withreference to the number of FDM carriers and their frequencyseparation. The combined MMSE-ML scheme achieves a nearoptimum error performance with polynomial complexity for asmall number of BPSK FDM carriers. For QPSK modulationthe performance of the proposed system improves for a largenumber of ML comparisons. In all cases, the detectability of theFDM signal is bounded by the signal dimension and the carriersfrequency distance.

I. INTRODUCTION

T He ever growing demand for broadband applications hasbeen a key driver for research into spectrally efficient

physical layer techniques that guarantee reliable transmissionand reception of high data rate signals. Orthogonal FrequencyDivision Multiplexing (OFDM) prevails in the area of wiredand wireless communications and is currently employed indiverse communication systems such as the Digital SubscriberLine (DSL), the Digital Audio and Video Broadcasting (DABand DVB, respectively), 802.11 a,g as well as the broadbandwireless access networks (WiMAX).

Recently, Rodrigues and Darwazeh [I] as well as Xiong [2]introduced systems that occupy half the OFDM bandwidth.However, the spectral efficiency benefit is not fully exploitedsince these systems can only operate with one-dimensionalmodulation (e.g. BPSK or M-ASK). Moreover, they appear tobe highly sensitive to wireless environments impairments suchas frequency offsets, timing offsets, and frequency selectivefading [3].

Additionally, Rodrigues and Darwazeh proposed a non-orthogonal Frequency Division Multiplexing (FDM) system[4]. This system exhibits higher spectral efficiency, at theexpense of higher detection complexity. For such systems itwas shown that ML detection is not computationally feasible

r(t) = s(t) + n(t) (2)

Input Stream SIP

b~(t)

c:.210.5(j) ~MSEWW(J)

~~

c.2co>.~

oC"0ooof:::lo.0.c.C)

"CDz

DC.5...J~ s=[11

Fig. 1. The combined MMSE-ML GS FDM system architecture.

It is appropriate to describe the process by a linear statisticalmodel

where R is the N x 1 vector matrix of sufficient statistics, Sis the N x 1 vector of transmitted symbols, M is the N x Nmatrix of the correlation coefficients between the FDM carriersand the orthonormal base, and N is a vector of N independentGaussian noise samples with zero mean and covariance matrixa 2IN (IN is the N x N identity matrix) [8]. Each of theelements of matrix M is given by

- {T j21r(a-l)b.. jtb* (t)dt f3 - 1 N (5)m a j3 - Jo

e (,6-1)' lX, - , ••. ,

The cross correlations matrix M is upper triangular [4], [9];effectively, the output of each correlator is contaminated byintercarrier interference. Optimal detection (in terms of errorperformance) is based on the maximum likelihood principle.The ML estimate can be expressed as follows [4], [8]

SML =~ {II(R - MS)1I2

} (6)

S

where 11·11 denotes the euclidean norm.It is important to note that ML detection is impractical,

because its computational complexity increases exponentiallywith the number of FDM carriers and the constellationcardinality. Therefore, we investigate the efficiency of twoalternatives: i) linear MMSE detection and ii) linear MMSEdetection combined with ML detection (see Fig. 1).

III. FDM SIGNAL DETECTION TECHNIQUES

A. MMSE Detection

In linear MMSE detection, the objective is to minimise themean square error between the vector of transmitted symbolsand the vector of their estimates, using solely linear operationsrepresented by the matrix GMMSE, Le.,

GMMSE = ~E{lls_sI12} (7)

G

~E{IIS-GRI12}G

R=MS+N, (4)

where E {.} is the expectation operator and M H denotes thehermitian of matrix M.

It is straightforward to show that the MMSE matrix is

GMMSE = MH (MMH + :; IN) -1 (8)

where a2 is the noise power and a; is the power of the FDMsignal (assuming transmitted symbols with a;IN covariancematrix.

Therefore, the MMSE estimate SM M S E is

SMMSE = MH (MMH + :;IN ) -1 R (9)We obtain the transmitted symbols estimates after slicing

SMMSE.

B. MMSE-ML Detection

In MMSE-ML procedure, we use a two step procedure. Ini-tially, we use an MMSE filter to generate an MMSE estimateSMMSE of the original transmitted symbols S. Subsequently,we use the ML principle in a neighborhood, V, of a slicedversion of MMSE estimate. The neighborhood consists ofthe set of transmitted symbols whose binary representationis within a certain Hamming distance, P, from the binaryrepresentation (sliced version) of the MMSE estimate. Thisprocedure is commonly known as boxed ML.

The neighborhood V consists of the set of transmittedvectors S obeying the relationship: dH(S', S~MSE) :::; pwhere dH ( " .) represents the Hamming distance operator, S'represents the binary version of S, and S~MSE represents thebinary version of SMMSE, i.e.,

S E V iff dH(S', S~MSE) :::; P (10)

C. MMSE-ML Complexity

The complexity of the proposed method depends on thenumber of calculations of both the MMSE and ML com-ponents of the algorithm. The former has a complexity ofpolynomial order O(N3) over the number of carriers N. Thecomputational cost of the latter depends on the size of theMMSE neighborhood V, since this determines the number of

TABLE IRATIO n OF ML OVER THE MMSE-ML COMPARISONS

BPSK BPSK QPSK QPSK

Carriers n,dH ==l n,dH==2 n,dH == 1 n,dH == 2

2 2 1.33 4 1.6

4 4 1.60 32 7.11

8 32 7.11 4096 481.88

16 4096 481.88 > 1010 > 8 X 106

It is apparent that for P equal to zero, the introducedscheme reduces to MMSE. For P equal to unity, the numberof necessary boxed ML comparisons is N log2 M, as opposedto the M N comparisons required for the ML implementationover the entire group of FDM symbols. Table I provides theratio Q of the number of ML over MMSE-ML comparisonsfor various FDM signal dimensions.

0.90.5 0.6 0.7 0.8Frequency Separation - dFT

0.410-3 '--_--L...__.L...-_---'-__..L...-_-....L.__...I..-_---l

0.3

.E0:::

~ 10-2

1D 10-1.

'0t.Ooz:0w

--cr-ZF-e-MMSE--&- MMSE-ML d

H=1

-+-- ML

gn(t) = ~ [gn(t) - ~ rgn(t)b;(t)dtbi(t)]V€n i=lio

end of n loop

end m loop (12)

where ~"~ is chosen so that the energy of the rnth orthonor-malised carrier is 1.

Simulation results show that the adaptation of the neworthonormalisation technique enables doubling the dimensionof the FDM signal without forcing the M matrix determinantto approach zero rapidly. Nevertheless, matrix singularityremains a fundamental limitation in signal detection of FDMsystems operating below the orthogonality limit.

IV. RESULTS

In our modelling we considered transmission over AWGNonly. We performed Bit Error Rate (BER) measurements ofsimulated systems for the proposed MMSE and MMSE-MLschemes for up to N == 48 FDM carriers with minimumfrequency separation equal to 0.3 of the inverse of the FDMsymbol period, ~f == 01'3, and a fixed value of Energy Per Bitto Noise Density Ratio (Eb/NO) equal to 5 dB. Carriers weremodulated either by real BPSK or complex QPSK symbols.In all simulations, ML and/or ZF detection curves are used asreference.

4 Carriers BPSK MGS FDM Detection Techniques Comparison10

0r----,---,..----,---...--;:::::==::r:==:==::::c===::::::;l

A. Error Performance

Figures 2 and 3 show BER versus the normalised carrierdistance (a fraction of the inverse of the FDM symbol period,dFT == ~f x T). Simulations were performed for a smallnumber (N == 4) of BPSK and QPSK FDM carriers. MMSE-ML measurements were taken with Hamming distance dH == 1for BPSK carriers and dH == {I, 2} for QPSK carriers.

(11)

size(D)

D. Modified Gram Schmidt

It is clear from section III that the inversion of matrixM is a prerequisite of the MMSE application. However, Meigenvalues (and consequently its determinant) go to zerorapidly with the increase of the number of carriers. Part of theproblem is due to the orthonormalisation procedure. ClassicGS (CGS) [4], [8] generates new vectors by subtracting fromthe respective initial carrier its projections (onto the alreadyorthonormalised carriers). Unfortunately, the projections op-eration introduces numerical errors that result in the processdegradation and consequently the loss of orthogonality after anumber of calculations. A significant improvement is the Mod-ified Gram Schmidt (MGS) method. While mathematicallyequivalent to CGS, MGS iterative nature results in a superiorcomputational performance [10]. The main difference fromthe classic GS is that while in CGS the algorithm calculatesthe m th vector and leaves the remaining m + 1 to N vectorsunaltered, in the modified version after the calculation of eachvector the remaining vectors are also transformed.

A simple implementation of the MGS is the following

the executed ML comparisons. The length of the expandedFDM symbols is equal to N x log2 M. Consequently, the sizeof the neighborhood V will be equal to the sum of all possiblecombinations of N x log2 M bits with k flipped bits taken ata time, where k runs from 1 to P, i.e.

pL (N10~2M)

k=lt (Nlog2 M)!k=l (Nlog2M - k)!k!

for rn from 1 to N

bm(t) = ~ [gm(t) - y:1

rT

gm (t)bi(t)dtbi(t)]V~m i=l io

for n from m + 1 to N

Fig. 2. BER of MMSE and MMSE-ML (dH == 1) detection versusnormalised frequency separation for N == 4 BPSK FDM carriers. Eb/Nois 5 dB.

Both figures show that MMSE is superior to ZF but infe-rior to ML for BPSK and QPSK FDM signals. MMSE-ML

653 4Eb/No (dB)

10-2 .LO,....aII~L1."C

.E0:::Wal 10-3 --ML

-----v-- 4 MMSE-ML--e- 8 MMSE-ML-+-24MMSE---.- 24 MMSE-ML-6---- 32 MMSE-ML--48 MMSE-ML

Error Performance for BPSK MMSE-ML MGS FDM10-1~---,.-----.---~-----,r___--..,....--_...__-__,

a small number of carriers. As the latter increases, a largerHamming distance (dH 2:: 2) is required to improve the singleMMSE detection.

0.9

-----v-- ZF--e-MMSE~ MMSE-ML d

H= 1

-A- MMSE-ML dH

= 2

~ML

0.5 0.6 0.7 0.8Frequency Separation - dFT

0.410-3 L.....-_----L.__---L-__....&....-._----'__--'-__-'--_---J

0.3

4 Carriers QPSK MGS FDM Detection Techniques Comparison10

0r-----r---,---,.-----,----;:===c====::r::::::==::::::;l

al 10-1

"CLO

oz:0w

Fig. 3. BER of MMSE and MMSE-ML detection versus normalisedfrequency separation for N == 4 QPSK FDM carriers. Eb/No is 5 dB andthe Hamming distance in MMSE-ML scheme is dH == {I, 2}.

653 4Eb/No (dB)

--ML-&-- 4MMSE-+- 4MMSE-ML d

H=1

~- 4MMSE-ML dH

=2

--e-8MMSE-6---- 8MMSE-ML d

H=1

~ 8MMSE-ML dH

=210-4 L::::==:=:r:====::r::===--.i......-_----i...__----i...-__..i.-_--.J

o

Error Performance for QPSK MMSE-ML MGS FDM100 r------r---...----~-__,.-----r---......-----.

0:::Wal

Fig. 6. BER of MMSE and MMSE-ML detection versus Eb/No for N =={2, 4,8, 16} QPSK FDM carriers. The carriers distance is dFT == 0;5 andthe Hamming distance in MMSE-ML scheme is dH == {I, 2}.

Fig. 5. BER of MMSE and MMSE-ML (dH == 1) detection versus Eb/Nofor N == ~4,8,24,36, 48} BPSK FDM carriers. The carriers distance isdFT == 015 .

0.9

-.-4ML~4MMSE-ML

--e- 8 MMSE-ML-A- 24 MMSE-ML----Iff- 32 MMSE-ML---.- 48 MMSE-ML

0.5 0.6 0.7 0.8Frequency Separation - dFT

BPSK MGS FDM MMSE - ML with dH

=1 Detection

0.410-3 L.....-_----L.__---L-__....&....-._----'__--'-__-'--_---J

0.3

0:::

~ 10-2

al 10-1

"CLO

oz:0w

performs close to ML for BPSK FDM carriers. However, theBER in the QPSK case depends on the selected Hammingdistance.

Further studies were performed for a larger number ofBPSK carriers N == 4, 8, 24, 36, 48 and an MMSE neighbor-hood with dH == 1. Fig. 4 shows that concatenating MMSE-ML detection introduces only a small error penalty as thenumber of BPSK FDM carriers increases.

Fig. 4. BER of MMSE-ML (dH == 1) detection versus normalised frequencyseparation for N == {4, 8, 24, 32, 48} FDM carriers. Eb/No is 5 dB.

In addition, we performed BER measurements for variousEb/No values, for a fixed dFT == 0.75. Fig. 5 shows thatBPSK performance almost matches the ML case. On the con-trary, Fig. 6 shows that in the QPSK case the error performancedepends on the length of the selected Hamming distance. FordH == 1, MMSE-ML performs worse than MMSE even for

B. Computational Complexity

To estimate the computational complexity of the proposedmethods, we measured the CPU execution time for differentsimulations scenarios. In all measurements, we consideredthe detection of 10 FDM symbols for Eb/No == 5 dB anddFT == 0.75. In addition, we evaluated MMSE-ML forHamming distances dH == {I, 2, 3}.

Figures 7 and 8 illustrate a comparison of indicative CPUtimes between ZF, MMSE and MMSE - ML detection. It is

BPSK MGS FDM Detection Techiques Complexity Comparison100 r;:::::::::c:=====:c====:::;r-----r------:-----.--~

MMSE and ML. This results in a much reduced complexityand might lead to feasible implementation. Standard MMSEdetection offers polynomial complexity at the expense of suboptimum error performance. To improve the latter, we appliedML in the neighborhood of the MMSE estimate. The orderof the MMSE-ML combinations was derived and simulationswere performed to evaluate its performance in AWGN. Thisnew scheme performance is close to that of optimum MLfor BPSK schemes at the expense of a small increase in thecomplexity. However, for QPSK the detection requires a largernumber of comparisons that increase significantly with thenumber of the carriers.

In addition, the detectability of the signal is restricted by thesignal dimension and the frequency separation that render theinversion of the cross correlations matrix impossible. Ongo-ing work includes investigating detection techniques such asSphere Decoding [11], a mathematical algorithm that promises

.ML performance with reduced complexity for complex FDMsignals.

//:."./..

. :/..................../ ..

/

60 .

40 .

20 .

50 .

30 .

10 .

---e-ZF90 -+-MMSEI --JV-- MMSE-ML dH=1 .

III 80 ----f:l-- MMSE-ML dH

=2 :' ..

~ 70 -&- MMSE-ML dH=3 :>-en~oLL.

~

.ECDEt=c:o~"SEen

clear that ZF and MMSE require less computational effort.MMSE - ML with dH equal to unity has simulation time com-parable to that ofMMSE . However, the order of its complexityincreases with the size of the MMSE neighborhood.

Fig. 7. CPU execution times for ZF, MMSE, and MMSE-ML FDM detectionfor 10 BPSK FDM symbols of N = 2 to 16 carriers. The carriers distanceis dFT = 0.75 of the inverse of the FDM symbol period T. MMSE-MLdetection is simulated for Hamming distance dH = {1, 2, 3}.

4 6 8 10 12Number of FDM carriers

14 16ACKNOWLEDGMENT

This work is supported by the UK Engineering andPhysical Sciences Research Council (EPSRC) under grantEP/D077362/1.

REFERENCES

[1] M. Rodrigues and I. Darwazeh, "Fast OFDM: A proposal for doublingthe data rate of OFDM schemes," in Proceedings of the InternationalConference on Telecommunications, Beijing, China, Jun. 2002, pp. 484-487.

[2] F. Xiong, "M-ary amplitude shift keying OFDM system," IEEE Trans-actions on Communications, vol. 51, no. 10, pp. 1638-1642, Oct. 2003.

[3] D. Karampatsis, M. Rodrigues, and I. Darwazeh, "Performance com-parison of OFDM and FOFDM communication systems in typicalmultipath environments: Effects of frequency selective fading, frequencyand timing offsets and I1Q QAM modulator/demodulator imbalance," in7th World Multiconference on Systemics, Cybernetics, and Informatics(SCI 2003), Orlando, USA, July 2003.

[4] M. Rodrigues and 1. Darwazeh, "A spectrally efficient frequency divisionmultiplexing based communication system," in 8th International OFDM-Workshop, Hamburg, Germany, Sep. 2003, pp. 70-74.

[5] A. 1. Paulraj, D. A. Gore, R. U. Nabar, and H. Blcskei, "An overview ofMIMO communications A key to gigabit wireless," IEEE Proceedings,vol. 92, no. 2, pp. 198-218, February 2004.

[6] V. Pammer, Y. Delignon, W. Sawaya, and D. Boulinguez, "A lowcomplexity suboptimal MIMO receiver: The combined ZF-MLD algo-rithm," 14th IEEE Proceedings on Personal, Indoor and Mobile RadioCommunications, 2003. PIMRC 2003, vol. 3, pp. 2271-2275 vol.3, 7-10Sept. 2003.

[7] F. Wang, Y. Xiong, and X. Yang, "Approximate ML detection based onMMSE for MIMO systems," PIERS Online, vol. 3, no. 4, pp. 475-480,2007.

[8] J. G. Proakis, Digital Communications 4th Edition. McGraw Hill, 2001.[9] L. L. Scharf, Statistical Signal Processing. Addison-Wesley, 1991.

[10] J. W. D. Ben Noble, Applied Linear Algebra, 2nd ed. Prentice Hall,1977.

[II] E. Viterbo and J. Boutros, "A universal lattice code decoder for fadingchannels," IEEE Transactions on Information Theory, vol. 45, no. 5, pp.1639-1642, Jul 1999.

16148 10 12Number of FDM carriers

64

10

.ECD 40Et=§ 30·~~ 20·en

-e-ZF90 -+-MMSE

I 80 --JV-- MMSE-ML dH=1

-5 ~ MMSE-ML dH=2! :: ~MMSE-ML dH=r

~ /o 50 /.

.!

100 ~=:::c======:::::c======::::;-r---.-----.----,--__,QPSK MGS FDM Detection Techiques Complexity

V. CONCLUSIONS AND FUTURE WORK

A new detection method for spectrally efficient FDM sys-tems has been proposed. The method is based on combining

Fig. 8. CPU execution times for ZF, MMSE, and MMSE-ML FDM detectionfor 10 QPSK FDM symbols of N = 2 to 16 carriers. The carriers distanceis dFT = 0.75 of the inverse of the FDM symbol period T. MMSE-MLdetection is simulated for Hamming distance dH = {I, 2, 3}.